deal.II version GIT relicensing-1721-g8100761196 2024-08-31 12:30:00+00:00
\(\newcommand{\dealvcentcolon}{\mathrel{\mathop{:}}}\) \(\newcommand{\dealcoloneq}{\dealvcentcolon\mathrel{\mkern-1.2mu}=}\) \(\newcommand{\jump}[1]{\left[\!\left[ #1 \right]\!\right]}\) \(\newcommand{\average}[1]{\left\{\!\left\{ #1 \right\}\!\right\}}\)
Loading...
Searching...
No Matches
step-75.h
Go to the documentation of this file.
1) const override
467 *   {
468 *   const std::array<double, 2> polar =
470 *  
471 *   constexpr const double alpha = 2. / 3.;
472 *   return std::pow(polar[0], alpha) * std::sin(alpha * polar[1]);
473 *   }
474 *   };
475 *  
476 *  
477 *  
478 * @endcode
479 *
480 *
481 * <a name="step_75-Parameters"></a>
482 * <h3>Parameters</h3>
483 *
484
485 *
486 * For this tutorial, we will use a simplified set of parameters. It is also
487 * possible to use a ParameterHandler class here, but to keep this tutorial
488 * short we decided on using simple structs. The actual intention of all these
489 * parameters will be described in the upcoming classes at their respective
490 * location where they are used.
491 *
492
493 *
494 * The following parameter set controls the coarse-grid solver, the smoothers,
495 * and the inter-grid transfer scheme of the multigrid mechanism.
496 * We populate it with default parameters.
497 *
498 * @code
499 *   struct MultigridParameters
500 *   {
501 *   struct
502 *   {
503 *   std::string type = "cg_with_amg";
504 *   unsigned int maxiter = 10000;
505 *   double abstol = 1e-20;
506 *   double reltol = 1e-4;
507 *   unsigned int smoother_sweeps = 1;
508 *   unsigned int n_cycles = 1;
509 *   std::string smoother_type = "ILU";
510 *   } coarse_solver;
511 *  
512 *   struct
513 *   {
514 *   std::string type = "chebyshev";
515 *   double smoothing_range = 20;
516 *   unsigned int degree = 5;
517 *   unsigned int eig_cg_n_iterations = 20;
518 *   } smoother;
519 *  
520 *   struct
521 *   {
523 *   p_sequence = MGTransferGlobalCoarseningTools::
524 *   PolynomialCoarseningSequenceType::decrease_by_one;
525 *   bool perform_h_transfer = true;
526 *   } transfer;
527 *   };
528 *  
529 *  
530 *  
531 * @endcode
532 *
533 * This is the general parameter struct for the problem class. You will find
534 * this struct divided into several categories, including general runtime
535 * parameters, level limits, refine and coarsen fractions, as well as
536 * parameters for cell weighting. It also contains an instance of the above
537 * struct for multigrid parameters which will be passed to the multigrid
538 * algorithm.
539 *
540 * @code
541 *   struct Parameters
542 *   {
543 *   unsigned int n_cycles = 8;
544 *   double tolerance_factor = 1e-12;
545 *  
546 *   MultigridParameters mg_data;
547 *  
548 *   unsigned int min_h_level = 5;
549 *   unsigned int max_h_level = 12;
550 *   unsigned int min_p_degree = 2;
551 *   unsigned int max_p_degree = 6;
552 *   unsigned int max_p_level_difference = 1;
553 *  
554 *   double refine_fraction = 0.3;
555 *   double coarsen_fraction = 0.03;
556 *   double p_refine_fraction = 0.9;
557 *   double p_coarsen_fraction = 0.9;
558 *  
559 *   double weighting_factor = 1.;
560 *   double weighting_exponent = 1.;
561 *   };
562 *  
563 *  
564 *  
565 * @endcode
566 *
567 *
568 * <a name="step_75-MatrixfreeLaplaceoperator"></a>
569 * <h3>Matrix-free Laplace operator</h3>
570 *
571
572 *
573 * This is a matrix-free implementation of the Laplace operator that will
574 * basically take over the part of the `assemble_system()` function from other
575 * tutorials. The meaning of all member functions will be explained at their
576 * definition later.
577 *
578
579 *
580 * We will use the FEEvaluation class to evaluate the solution vector
581 * at the quadrature points and to perform the integration. In contrast to
582 * other tutorials, the template arguments `degree` is set to @f$-1@f$ and
583 * `number of quadrature in 1d` to @f$0@f$. In this case, FEEvaluation selects
584 * dynamically the correct polynomial degree and number of quadrature
585 * points. Here, we introduce an alias to FEEvaluation with the correct
586 * template parameters so that we do not have to worry about them later on.
587 *
588 * @code
589 *   template <int dim, typename number>
590 *   class LaplaceOperator : public Subscriptor
591 *   {
592 *   public:
594 *  
595 *   using FECellIntegrator = FEEvaluation<dim, -1, 0, 1, number>;
596 *  
597 *   LaplaceOperator() = default;
598 *  
599 *   LaplaceOperator(const hp::MappingCollection<dim> &mapping,
600 *   const DoFHandler<dim> &dof_handler,
601 *   const hp::QCollection<dim> &quad,
602 *   const AffineConstraints<number> &constraints,
603 *   VectorType &system_rhs);
604 *  
605 *   void reinit(const hp::MappingCollection<dim> &mapping,
606 *   const DoFHandler<dim> &dof_handler,
607 *   const hp::QCollection<dim> &quad,
608 *   const AffineConstraints<number> &constraints,
609 *   VectorType &system_rhs);
610 *  
611 *   types::global_dof_index m() const;
612 *  
613 *   number el(unsigned int, unsigned int) const;
614 *  
615 *   void initialize_dof_vector(VectorType &vec) const;
616 *  
617 *   void vmult(VectorType &dst, const VectorType &src) const;
618 *  
619 *   void Tvmult(VectorType &dst, const VectorType &src) const;
620 *  
621 *   const TrilinosWrappers::SparseMatrix &get_system_matrix() const;
622 *  
623 *   void compute_inverse_diagonal(VectorType &diagonal) const;
624 *  
625 *   private:
626 *   void do_cell_integral_local(FECellIntegrator &integrator) const;
627 *  
628 *   void do_cell_integral_global(FECellIntegrator &integrator,
629 *   VectorType &dst,
630 *   const VectorType &src) const;
631 *  
632 *  
633 *   void do_cell_integral_range(
634 *   const MatrixFree<dim, number> &matrix_free,
635 *   VectorType &dst,
636 *   const VectorType &src,
637 *   const std::pair<unsigned int, unsigned int> &range) const;
638 *  
639 *   MatrixFree<dim, number> matrix_free;
640 *  
641 * @endcode
642 *
643 * To solve the equation system on the coarsest level with an AMG
644 * preconditioner, we need an actual system matrix on the coarsest level.
645 * For this purpose, we provide a mechanism that optionally computes a
646 * matrix from the matrix-free formulation, for which we introduce a
647 * dedicated SparseMatrix object. In the default case, this matrix stays
648 * empty. Once `get_system_matrix()` is called, this matrix is filled (lazy
649 * allocation). Since this is a `const` function, we need the "mutable"
650 * keyword here. We also need a the constraints object to build the matrix.
651 *
652 * @code
653 *   AffineConstraints<number> constraints;
654 *   mutable TrilinosWrappers::SparseMatrix system_matrix;
655 *   };
656 *  
657 *  
658 *  
659 * @endcode
660 *
661 * The following section contains functions to initialize and reinitialize
662 * the class. In particular, these functions initialize the internal
663 * MatrixFree instance. For sake of simplicity, we also compute the system
664 * right-hand-side vector.
665 *
666 * @code
667 *   template <int dim, typename number>
668 *   LaplaceOperator<dim, number>::LaplaceOperator(
669 *   const hp::MappingCollection<dim> &mapping,
670 *   const DoFHandler<dim> &dof_handler,
671 *   const hp::QCollection<dim> &quad,
672 *   const AffineConstraints<number> &constraints,
673 *   VectorType &system_rhs)
674 *   {
675 *   this->reinit(mapping, dof_handler, quad, constraints, system_rhs);
676 *   }
677 *  
678 *  
679 *  
680 *   template <int dim, typename number>
681 *   void LaplaceOperator<dim, number>::reinit(
682 *   const hp::MappingCollection<dim> &mapping,
683 *   const DoFHandler<dim> &dof_handler,
684 *   const hp::QCollection<dim> &quad,
685 *   const AffineConstraints<number> &constraints,
686 *   VectorType &system_rhs)
687 *   {
688 * @endcode
689 *
690 * Clear internal data structures (in the case that the operator is reused).
691 *
692 * @code
693 *   this->system_matrix.clear();
694 *  
695 * @endcode
696 *
697 * Copy the constraints, since they might be needed for computation of the
698 * system matrix later on.
699 *
700 * @code
701 *   this->constraints.copy_from(constraints);
702 *  
703 * @endcode
704 *
705 * Set up MatrixFree. At the quadrature points, we only need to evaluate
706 * the gradient of the solution and test with the gradient of the shape
707 * functions so that we only need to set the flag `update_gradients`.
708 *
709 * @code
712 *  
713 *   matrix_free.reinit(mapping, dof_handler, constraints, quad, data);
714 *  
715 * @endcode
716 *
717 * Compute the right-hand side vector. For this purpose, we set up a second
718 * MatrixFree instance that uses a modified AffineConstraints not containing
719 * the constraints due to Dirichlet-boundary conditions. This modified
720 * operator is applied to a vector with only the Dirichlet values set. The
721 * result is the negative right-hand-side vector.
722 *
723 * @code
724 *   {
725 *   AffineConstraints<number> constraints_without_dbc(
726 *   dof_handler.locally_owned_dofs(),
728 *  
730 *   constraints_without_dbc);
731 *   constraints_without_dbc.close();
732 *  
733 *   VectorType b, x;
734 *  
735 *   this->initialize_dof_vector(system_rhs);
736 *  
737 *   MatrixFree<dim, number> matrix_free;
738 *   matrix_free.reinit(
739 *   mapping, dof_handler, constraints_without_dbc, quad, data);
740 *  
741 *   matrix_free.initialize_dof_vector(b);
742 *   matrix_free.initialize_dof_vector(x);
743 *  
744 *   constraints.distribute(x);
745 *  
746 *   matrix_free.cell_loop(&LaplaceOperator::do_cell_integral_range,
747 *   this,
748 *   b,
749 *   x);
750 *  
751 *   constraints.set_zero(b);
752 *  
753 *   system_rhs -= b;
754 *   }
755 *   }
756 *  
757 *  
758 *  
759 * @endcode
760 *
761 * The following functions are implicitly needed by the multigrid algorithm,
762 * including the smoothers.
763 *
764
765 *
766 * Since we do not have a matrix, query the DoFHandler for the number of
767 * degrees of freedom.
768 *
769 * @code
770 *   template <int dim, typename number>
771 *   types::global_dof_index LaplaceOperator<dim, number>::m() const
772 *   {
773 *   return matrix_free.get_dof_handler().n_dofs();
774 *   }
775 *  
776 *  
777 *  
778 * @endcode
779 *
780 * Access a particular element in the matrix. This function is neither
781 * needed nor implemented, however, is required to compile the program.
782 *
783 * @code
784 *   template <int dim, typename number>
785 *   number LaplaceOperator<dim, number>::el(unsigned int, unsigned int) const
786 *   {
788 *   return 0;
789 *   }
790 *  
791 *  
792 *  
793 * @endcode
794 *
795 * Initialize the given vector. We simply delegate the task to the
796 * MatrixFree function with the same name.
797 *
798 * @code
799 *   template <int dim, typename number>
800 *   void
801 *   LaplaceOperator<dim, number>::initialize_dof_vector(VectorType &vec) const
802 *   {
803 *   matrix_free.initialize_dof_vector(vec);
804 *   }
805 *  
806 *  
807 *  
808 * @endcode
809 *
810 * Perform an operator evaluation by looping with the help of MatrixFree
811 * over all cells and evaluating the effect of the cell integrals (see also:
812 * `do_cell_integral_local()` and `do_cell_integral_global()`).
813 *
814 * @code
815 *   template <int dim, typename number>
816 *   void LaplaceOperator<dim, number>::vmult(VectorType &dst,
817 *   const VectorType &src) const
818 *   {
819 *   this->matrix_free.cell_loop(
820 *   &LaplaceOperator::do_cell_integral_range, this, dst, src, true);
821 *   }
822 *  
823 *  
824 *  
825 * @endcode
826 *
827 * Perform the transposed operator evaluation. Since we are considering
828 * symmetric "matrices", this function can simply delegate it task to vmult().
829 *
830 * @code
831 *   template <int dim, typename number>
832 *   void LaplaceOperator<dim, number>::Tvmult(VectorType &dst,
833 *   const VectorType &src) const
834 *   {
835 *   this->vmult(dst, src);
836 *   }
837 *  
838 *  
839 *  
840 * @endcode
841 *
842 * Since we do not have a system matrix, we cannot loop over the the
843 * diagonal entries of the matrix. Instead, we compute the diagonal by
844 * performing a sequence of operator evaluations to unit basis vectors.
845 * For this purpose, an optimized function from the MatrixFreeTools
846 * namespace is used. The inversion is performed manually afterwards.
847 *
848 * @code
849 *   template <int dim, typename number>
850 *   void LaplaceOperator<dim, number>::compute_inverse_diagonal(
851 *   VectorType &diagonal) const
852 *   {
853 *   this->matrix_free.initialize_dof_vector(diagonal);
854 *   MatrixFreeTools::compute_diagonal(matrix_free,
855 *   diagonal,
856 *   &LaplaceOperator::do_cell_integral_local,
857 *   this);
858 *  
859 *   for (auto &i : diagonal)
860 *   i = (std::abs(i) > 1.0e-10) ? (1.0 / i) : 1.0;
861 *   }
862 *  
863 *  
864 *  
865 * @endcode
866 *
867 * In the matrix-free context, no system matrix is set up during
868 * initialization of this class. As a consequence, it has to be computed
869 * here if it should be requested. Since the matrix is only computed in
870 * this tutorial for linear elements (on the coarse grid), this is
871 * acceptable.
872 * The matrix entries are obtained via sequence of operator evaluations.
873 * For this purpose, the optimized function MatrixFreeTools::compute_matrix()
874 * is used. The matrix will only be computed if it has not been set up yet
875 * (lazy allocation).
876 *
877 * @code
878 *   template <int dim, typename number>
880 *   LaplaceOperator<dim, number>::get_system_matrix() const
881 *   {
882 *   if (system_matrix.m() == 0 && system_matrix.n() == 0)
883 *   {
884 *   const auto &dof_handler = this->matrix_free.get_dof_handler();
885 *  
887 *   dof_handler.locally_owned_dofs(),
888 *   dof_handler.get_triangulation().get_mpi_communicator());
889 *  
890 *   DoFTools::make_sparsity_pattern(dof_handler, dsp, this->constraints);
891 *  
892 *   dsp.compress();
893 *   system_matrix.reinit(dsp);
894 *  
896 *   matrix_free,
897 *   constraints,
898 *   system_matrix,
899 *   &LaplaceOperator::do_cell_integral_local,
900 *   this);
901 *   }
902 *  
903 *   return this->system_matrix;
904 *   }
905 *  
906 *  
907 *  
908 * @endcode
909 *
910 * Perform cell integral on a cell batch without gathering and scattering
911 * the values. This function is needed for the MatrixFreeTools functions
912 * since these functions operate directly on the buffers of FEEvaluation.
913 *
914 * @code
915 *   template <int dim, typename number>
916 *   void LaplaceOperator<dim, number>::do_cell_integral_local(
917 *   FECellIntegrator &integrator) const
918 *   {
919 *   integrator.evaluate(EvaluationFlags::gradients);
920 *  
921 *   for (const unsigned int q : integrator.quadrature_point_indices())
922 *   integrator.submit_gradient(integrator.get_gradient(q), q);
923 *  
924 *   integrator.integrate(EvaluationFlags::gradients);
925 *   }
926 *  
927 *  
928 *  
929 * @endcode
930 *
931 * Same as above but with access to the global vectors.
932 *
933 * @code
934 *   template <int dim, typename number>
935 *   void LaplaceOperator<dim, number>::do_cell_integral_global(
936 *   FECellIntegrator &integrator,
937 *   VectorType &dst,
938 *   const VectorType &src) const
939 *   {
940 *   integrator.gather_evaluate(src, EvaluationFlags::gradients);
941 *  
942 *   for (const unsigned int q : integrator.quadrature_point_indices())
943 *   integrator.submit_gradient(integrator.get_gradient(q), q);
944 *  
945 *   integrator.integrate_scatter(EvaluationFlags::gradients, dst);
946 *   }
947 *  
948 *  
949 *  
950 * @endcode
951 *
952 * This function loops over all cell batches within a cell-batch range and
953 * calls the above function.
954 *
955 * @code
956 *   template <int dim, typename number>
957 *   void LaplaceOperator<dim, number>::do_cell_integral_range(
958 *   const MatrixFree<dim, number> &matrix_free,
959 *   VectorType &dst,
960 *   const VectorType &src,
961 *   const std::pair<unsigned int, unsigned int> &range) const
962 *   {
963 *   FECellIntegrator integrator(matrix_free, range);
964 *  
965 *   for (unsigned cell = range.first; cell < range.second; ++cell)
966 *   {
967 *   integrator.reinit(cell);
968 *  
969 *   do_cell_integral_global(integrator, dst, src);
970 *   }
971 *   }
972 *  
973 *  
974 *  
975 * @endcode
976 *
977 *
978 * <a name="step_75-Solverandpreconditioner"></a>
979 * <h3>Solver and preconditioner</h3>
980 *
981
982 *
983 *
984 * <a name="step_75-Conjugategradientsolverwithmultigridpreconditioner"></a>
985 * <h4>Conjugate-gradient solver with multigrid preconditioner</h4>
986 *
987
988 *
989 * This function solves the equation system with a sequence of provided
990 * multigrid objects. It is meant to be treated as general as possible, hence
991 * the multitude of template parameters.
992 *
993 * @code
994 *   template <typename VectorType,
995 *   int dim,
996 *   typename SystemMatrixType,
997 *   typename LevelMatrixType,
998 *   typename MGTransferType>
999 *   static void
1000 *   mg_solve(SolverControl &solver_control,
1001 *   VectorType &dst,
1002 *   const VectorType &src,
1003 *   const MultigridParameters &mg_data,
1004 *   const DoFHandler<dim> &dof,
1005 *   const SystemMatrixType &fine_matrix,
1006 *   const MGLevelObject<std::unique_ptr<LevelMatrixType>> &mg_matrices,
1007 *   const MGTransferType &mg_transfer)
1008 *   {
1009 *   AssertThrow(mg_data.coarse_solver.type == "cg_with_amg",
1010 *   ExcNotImplemented());
1011 *   AssertThrow(mg_data.smoother.type == "chebyshev", ExcNotImplemented());
1012 *  
1013 *   const unsigned int min_level = mg_matrices.min_level();
1014 *   const unsigned int max_level = mg_matrices.max_level();
1015 *  
1016 *   using SmootherPreconditionerType = DiagonalMatrix<VectorType>;
1017 *   using SmootherType = PreconditionChebyshev<LevelMatrixType,
1018 *   VectorType,
1019 *   SmootherPreconditionerType>;
1020 *   using PreconditionerType = PreconditionMG<dim, VectorType, MGTransferType>;
1021 *  
1022 * @endcode
1023 *
1024 * We initialize level operators and Chebyshev smoothers here.
1025 *
1026 * @code
1027 *   mg::Matrix<VectorType> mg_matrix(mg_matrices);
1028 *  
1030 *   min_level, max_level);
1031 *  
1032 *   for (unsigned int level = min_level; level <= max_level; ++level)
1033 *   {
1034 *   smoother_data[level].preconditioner =
1035 *   std::make_shared<SmootherPreconditionerType>();
1036 *   mg_matrices[level]->compute_inverse_diagonal(
1037 *   smoother_data[level].preconditioner->get_vector());
1038 *   smoother_data[level].smoothing_range = mg_data.smoother.smoothing_range;
1039 *   smoother_data[level].degree = mg_data.smoother.degree;
1040 *   smoother_data[level].eig_cg_n_iterations =
1041 *   mg_data.smoother.eig_cg_n_iterations;
1042 *   }
1043 *  
1045 *   mg_smoother;
1046 *   mg_smoother.initialize(mg_matrices, smoother_data);
1047 *  
1048 * @endcode
1049 *
1050 * Next, we initialize the coarse-grid solver. We use conjugate-gradient
1051 * method with AMG as preconditioner.
1052 *
1053 * @code
1054 *   ReductionControl coarse_grid_solver_control(mg_data.coarse_solver.maxiter,
1055 *   mg_data.coarse_solver.abstol,
1056 *   mg_data.coarse_solver.reltol,
1057 *   false,
1058 *   false);
1059 *   SolverCG<VectorType> coarse_grid_solver(coarse_grid_solver_control);
1060 *  
1061 *   std::unique_ptr<MGCoarseGridBase<VectorType>> mg_coarse;
1062 *  
1063 *   TrilinosWrappers::PreconditionAMG precondition_amg;
1065 *   amg_data.smoother_sweeps = mg_data.coarse_solver.smoother_sweeps;
1066 *   amg_data.n_cycles = mg_data.coarse_solver.n_cycles;
1067 *   amg_data.smoother_type = mg_data.coarse_solver.smoother_type.c_str();
1068 *  
1069 *   precondition_amg.initialize(mg_matrices[min_level]->get_system_matrix(),
1070 *   amg_data);
1071 *  
1072 *   mg_coarse =
1073 *   std::make_unique<MGCoarseGridIterativeSolver<VectorType,
1075 *   LevelMatrixType,
1076 *   decltype(precondition_amg)>>(
1077 *   coarse_grid_solver, *mg_matrices[min_level], precondition_amg);
1078 *  
1079 * @endcode
1080 *
1081 * Finally, we create the Multigrid object, convert it to a preconditioner,
1082 * and use it inside of a conjugate-gradient solver to solve the linear
1083 * system of equations.
1084 *
1085 * @code
1087 *   mg_matrix, *mg_coarse, mg_transfer, mg_smoother, mg_smoother);
1088 *  
1089 *   PreconditionerType preconditioner(dof, mg, mg_transfer);
1090 *  
1091 *   SolverCG<VectorType>(solver_control)
1092 *   .solve(fine_matrix, dst, src, preconditioner);
1093 *   }
1094 *  
1095 *  
1096 *  
1097 * @endcode
1098 *
1099 *
1100 * <a name="step_75-Hybridpolynomialgeometricglobalcoarseningmultigridpreconditioner"></a>
1101 * <h4>Hybrid polynomial/geometric-global-coarsening multigrid preconditioner</h4>
1102 *
1103
1104 *
1105 * The above function deals with the actual solution for a given sequence of
1106 * multigrid objects. This functions creates the actual multigrid levels, in
1107 * particular the operators, and the transfer operator as a
1109 *
1110 * @code
1111 *   template <typename VectorType, typename OperatorType, int dim>
1112 *   void solve_with_gmg(SolverControl &solver_control,
1113 *   const OperatorType &system_matrix,
1114 *   VectorType &dst,
1115 *   const VectorType &src,
1116 *   const MultigridParameters &mg_data,
1117 *   const hp::MappingCollection<dim> &mapping_collection,
1118 *   const DoFHandler<dim> &dof_handler,
1119 *   const hp::QCollection<dim> &quadrature_collection)
1120 *   {
1121 * @endcode
1122 *
1123 * Create a DoFHandler and operator for each multigrid level,
1124 * as well as, create transfer operators. To be able to
1125 * set up the operators, we need a set of DoFHandler that we create
1126 * via global coarsening of p or h. For latter, we need also a sequence
1127 * of Triangulation objects that are obtained by
1129 *
1130
1131 *
1132 * In case no h-transfer is requested, we provide an empty deleter for the
1133 * `emplace_back()` function, since the Triangulation of our DoFHandler is
1134 * an external field and its destructor is called somewhere else.
1135 *
1136 * @code
1137 *   MGLevelObject<DoFHandler<dim>> dof_handlers;
1140 *  
1141 *   std::vector<std::shared_ptr<const Triangulation<dim>>>
1142 *   coarse_grid_triangulations;
1143 *   if (mg_data.transfer.perform_h_transfer)
1144 *   coarse_grid_triangulations =
1146 *   dof_handler.get_triangulation());
1147 *   else
1148 *   coarse_grid_triangulations.emplace_back(
1149 *   &(dof_handler.get_triangulation()), [](auto *) {});
1150 *  
1151 * @endcode
1152 *
1153 * Determine the total number of levels for the multigrid operation and
1154 * allocate sufficient memory for all levels.
1155 *
1156 * @code
1157 *   const unsigned int n_h_levels = coarse_grid_triangulations.size() - 1;
1158 *  
1159 *   const auto get_max_active_fe_degree = [&](const auto &dof_handler) {
1160 *   unsigned int max = 0;
1161 *  
1162 *   for (auto &cell : dof_handler.active_cell_iterators())
1163 *   if (cell->is_locally_owned())
1164 *   max =
1165 *   std::max(max, dof_handler.get_fe(cell->active_fe_index()).degree);
1166 *  
1167 *   return Utilities::MPI::max(max, MPI_COMM_WORLD);
1168 *   };
1169 *  
1170 *   const unsigned int n_p_levels =
1172 *   get_max_active_fe_degree(dof_handler), mg_data.transfer.p_sequence)
1173 *   .size();
1174 *  
1175 *   std::map<unsigned int, unsigned int> fe_index_for_degree;
1176 *   for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i)
1177 *   {
1178 *   const unsigned int degree = dof_handler.get_fe(i).degree;
1179 *   Assert(fe_index_for_degree.find(degree) == fe_index_for_degree.end(),
1180 *   ExcMessage("FECollection does not contain unique degrees."));
1181 *   fe_index_for_degree[degree] = i;
1182 *   }
1183 *  
1184 *   unsigned int minlevel = 0;
1185 *   unsigned int maxlevel = n_h_levels + n_p_levels - 1;
1186 *  
1187 *   dof_handlers.resize(minlevel, maxlevel);
1188 *   operators.resize(minlevel, maxlevel);
1189 *   transfers.resize(minlevel, maxlevel);
1190 *  
1191 * @endcode
1192 *
1193 * Loop from the minimum (coarsest) to the maximum (finest) level and set up
1194 * DoFHandler accordingly. We start with the h-levels, where we distribute
1195 * on increasingly finer meshes linear elements.
1196 *
1197 * @code
1198 *   for (unsigned int l = 0; l < n_h_levels; ++l)
1199 *   {
1200 *   dof_handlers[l].reinit(*coarse_grid_triangulations[l]);
1201 *   dof_handlers[l].distribute_dofs(dof_handler.get_fe_collection());
1202 *   }
1203 *  
1204 * @endcode
1205 *
1206 * After we reached the finest mesh, we will adjust the polynomial degrees
1207 * on each level. We reverse iterate over our data structure and start at
1208 * the finest mesh that contains all information about the active FE
1209 * indices. We then lower the polynomial degree of each cell level by level.
1210 *
1211 * @code
1212 *   for (unsigned int i = 0, l = maxlevel; i < n_p_levels; ++i, --l)
1213 *   {
1214 *   dof_handlers[l].reinit(dof_handler.get_triangulation());
1215 *  
1216 *   if (l == maxlevel) // finest level
1217 *   {
1218 *   auto &dof_handler_mg = dof_handlers[l];
1219 *  
1220 *   auto cell_other = dof_handler.begin_active();
1221 *   for (auto &cell : dof_handler_mg.active_cell_iterators())
1222 *   {
1223 *   if (cell->is_locally_owned())
1224 *   cell->set_active_fe_index(cell_other->active_fe_index());
1225 *   ++cell_other;
1226 *   }
1227 *   }
1228 *   else // coarse level
1229 *   {
1230 *   auto &dof_handler_fine = dof_handlers[l + 1];
1231 *   auto &dof_handler_coarse = dof_handlers[l + 0];
1232 *  
1233 *   auto cell_other = dof_handler_fine.begin_active();
1234 *   for (auto &cell : dof_handler_coarse.active_cell_iterators())
1235 *   {
1236 *   if (cell->is_locally_owned())
1237 *   {
1238 *   const unsigned int next_degree =
1241 *   cell_other->get_fe().degree,
1242 *   mg_data.transfer.p_sequence);
1243 *   Assert(fe_index_for_degree.find(next_degree) !=
1244 *   fe_index_for_degree.end(),
1245 *   ExcMessage("Next polynomial degree in sequence "
1246 *   "does not exist in FECollection."));
1247 *  
1248 *   cell->set_active_fe_index(fe_index_for_degree[next_degree]);
1249 *   }
1250 *   ++cell_other;
1251 *   }
1252 *   }
1253 *  
1254 *   dof_handlers[l].distribute_dofs(dof_handler.get_fe_collection());
1255 *   }
1256 *  
1257 * @endcode
1258 *
1259 * Next, we will create all data structures additionally needed on each
1260 * multigrid level. This involves determining constraints with homogeneous
1261 * Dirichlet boundary conditions, and building the operator just like on the
1262 * active level.
1263 *
1264 * @code
1266 *   constraints(minlevel, maxlevel);
1267 *  
1268 *   for (unsigned int level = minlevel; level <= maxlevel; ++level)
1269 *   {
1270 *   const auto &dof_handler = dof_handlers[level];
1271 *   auto &constraint = constraints[level];
1272 *  
1273 *   constraint.reinit(dof_handler.locally_owned_dofs(),
1274 *   DoFTools::extract_locally_relevant_dofs(dof_handler));
1275 *  
1276 *   DoFTools::make_hanging_node_constraints(dof_handler, constraint);
1277 *   VectorTools::interpolate_boundary_values(mapping_collection,
1278 *   dof_handler,
1279 *   0,
1281 *   constraint);
1282 *   constraint.close();
1283 *  
1284 *   VectorType dummy;
1285 *  
1286 *   operators[level] = std::make_unique<OperatorType>(mapping_collection,
1287 *   dof_handler,
1288 *   quadrature_collection,
1289 *   constraint,
1290 *   dummy);
1291 *   }
1292 *  
1293 * @endcode
1294 *
1295 * Set up intergrid operators and collect transfer operators within a single
1296 * operator as needed by the Multigrid solver class.
1297 *
1298 * @code
1299 *   for (unsigned int level = minlevel; level < maxlevel; ++level)
1300 *   transfers[level + 1].reinit(dof_handlers[level + 1],
1301 *   dof_handlers[level],
1302 *   constraints[level + 1],
1303 *   constraints[level]);
1304 *  
1306 *   transfers, [&](const auto l, auto &vec) {
1307 *   operators[l]->initialize_dof_vector(vec);
1308 *   });
1309 *  
1310 * @endcode
1311 *
1312 * Finally, proceed to solve the problem with multigrid.
1313 *
1314 * @code
1315 *   mg_solve(solver_control,
1316 *   dst,
1317 *   src,
1318 *   mg_data,
1319 *   dof_handler,
1320 *   system_matrix,
1321 *   operators,
1322 *   transfer);
1323 *   }
1324 *  
1325 *  
1326 *  
1327 * @endcode
1328 *
1329 *
1330 * <a name="step_75-ThecodeLaplaceProblemcodeclasstemplate"></a>
1331 * <h3>The <code>LaplaceProblem</code> class template</h3>
1332 *
1333
1334 *
1335 * Now we will finally declare the main class of this program, which solves
1336 * the Laplace equation on subsequently refined function spaces. Its structure
1337 * will look familiar as it is similar to the main classes of @ref step_27 "step-27" and
1338 * @ref step_40 "step-40". There are basically just two additions:
1339 * - The SparseMatrix object that would hold the system matrix has been
1340 * replaced by an object of the LaplaceOperator class for the MatrixFree
1341 * formulation.
1342 * - An object of parallel::CellWeights, which will help us with load
1343 * balancing, has been added.
1344 *
1345 * @code
1346 *   template <int dim>
1347 *   class LaplaceProblem
1348 *   {
1349 *   public:
1350 *   LaplaceProblem(const Parameters &parameters);
1351 *  
1352 *   void run();
1353 *  
1354 *   private:
1355 *   void initialize_grid();
1356 *   void setup_system();
1357 *   void print_diagnostics();
1358 *   void solve_system();
1359 *   void compute_indicators();
1360 *   void adapt_resolution();
1361 *   void output_results(const unsigned int cycle);
1362 *  
1363 *   MPI_Comm mpi_communicator;
1364 *  
1365 *   const Parameters prm;
1366 *  
1368 *   DoFHandler<dim> dof_handler;
1369 *  
1370 *   hp::MappingCollection<dim> mapping_collection;
1371 *   hp::FECollection<dim> fe_collection;
1372 *   hp::QCollection<dim> quadrature_collection;
1373 *   hp::QCollection<dim - 1> face_quadrature_collection;
1374 *  
1375 *   IndexSet locally_owned_dofs;
1376 *   IndexSet locally_relevant_dofs;
1377 *  
1378 *   AffineConstraints<double> constraints;
1379 *  
1380 *   LaplaceOperator<dim, double> laplace_operator;
1381 *   LinearAlgebra::distributed::Vector<double> locally_relevant_solution;
1383 *  
1384 *   std::unique_ptr<FESeries::Legendre<dim>> legendre;
1385 *   std::unique_ptr<parallel::CellWeights<dim>> cell_weights;
1386 *  
1387 *   Vector<float> estimated_error_per_cell;
1388 *   Vector<float> hp_decision_indicators;
1389 *  
1390 *   ConditionalOStream pcout;
1391 *   TimerOutput computing_timer;
1392 *   };
1393 *  
1394 *  
1395 *  
1396 * @endcode
1397 *
1398 *
1399 * <a name="step_75-ThecodeLaplaceProblemcodeclassimplementation"></a>
1400 * <h3>The <code>LaplaceProblem</code> class implementation</h3>
1401 *
1402
1403 *
1404 *
1405 * <a name="step_75-Constructor"></a>
1406 * <h4>Constructor</h4>
1407 *
1408
1409 *
1410 * The constructor starts with an initializer list that looks similar to the
1411 * one of @ref step_40 "step-40". We again prepare the ConditionalOStream object to allow
1412 * only the first process to output anything over the console, and initialize
1413 * the computing timer properly.
1414 *
1415 * @code
1416 *   template <int dim>
1417 *   LaplaceProblem<dim>::LaplaceProblem(const Parameters &parameters)
1418 *   : mpi_communicator(MPI_COMM_WORLD)
1419 *   , prm(parameters)
1420 *   , triangulation(mpi_communicator)
1421 *   , dof_handler(triangulation)
1422 *   , pcout(std::cout,
1423 *   (Utilities::MPI::this_mpi_process(mpi_communicator) == 0))
1424 *   , computing_timer(mpi_communicator,
1425 *   pcout,
1426 *   TimerOutput::never,
1428 *   {
1429 *   Assert(prm.min_h_level <= prm.max_h_level,
1430 *   ExcMessage(
1431 *   "Triangulation level limits have been incorrectly set up."));
1432 *   Assert(prm.min_p_degree <= prm.max_p_degree,
1433 *   ExcMessage("FECollection degrees have been incorrectly set up."));
1434 *  
1435 * @endcode
1436 *
1437 * We need to prepare the data structures for the hp-functionality in the
1438 * actual body of the constructor, and create corresponding objects for
1439 * every degree in the specified range from the parameter struct. As we are
1440 * only dealing with non-distorted rectangular cells, a linear mapping
1441 * object is sufficient in this context.
1442 *
1443
1444 *
1445 * In the Parameters struct, we provide ranges for levels on which the
1446 * function space is operating with a reasonable resolution. The multigrid
1447 * algorithm requires linear elements on the coarsest possible level. So we
1448 * start with the lowest polynomial degree and fill the collection with
1449 * consecutively higher degrees until the user-specified maximum is
1450 * reached.
1451 *
1452 * @code
1453 *   mapping_collection.push_back(MappingQ1<dim>());
1454 *  
1455 *   for (unsigned int degree = 1; degree <= prm.max_p_degree; ++degree)
1456 *   {
1457 *   fe_collection.push_back(FE_Q<dim>(degree));
1458 *   quadrature_collection.push_back(QGauss<dim>(degree + 1));
1459 *   face_quadrature_collection.push_back(QGauss<dim - 1>(degree + 1));
1460 *   }
1461 *  
1462 * @endcode
1463 *
1464 * As our FECollection contains more finite elements than we want to use for
1465 * the finite element approximation of our solution, we would like to limit
1466 * the range on which active FE indices can operate on. For this, the
1467 * FECollection class allows to register a hierarchy that determines the
1468 * succeeding and preceding finite element in case of of p-refinement and
1469 * p-coarsening, respectively. All functions in the hp::Refinement namespace
1470 * consult this hierarchy to determine future FE indices. We will register
1471 * such a hierarchy that only works on finite elements with polynomial
1472 * degrees in the proposed range <code>[min_p_degree, max_p_degree]</code>.
1473 *
1474 * @code
1475 *   const unsigned int min_fe_index = prm.min_p_degree - 1;
1476 *   fe_collection.set_hierarchy(
1477 *   /*next_index=*/
1478 *   [](const typename hp::FECollection<dim> &fe_collection,
1479 *   const unsigned int fe_index) -> unsigned int {
1480 *   return ((fe_index + 1) < fe_collection.size()) ? fe_index + 1 :
1481 *   fe_index;
1482 *   },
1483 *   /*previous_index=*/
1484 *   [min_fe_index](const typename hp::FECollection<dim> &,
1485 *   const unsigned int fe_index) -> unsigned int {
1486 *   Assert(fe_index >= min_fe_index,
1487 *   ExcMessage("Finite element is not part of hierarchy!"));
1488 *   return (fe_index > min_fe_index) ? fe_index - 1 : fe_index;
1489 *   });
1490 *  
1491 * @endcode
1492 *
1493 * We initialize the FESeries::Legendre object in the default configuration
1494 * for smoothness estimation.
1495 *
1496 * @code
1497 *   legendre = std::make_unique<FESeries::Legendre<dim>>(
1499 *  
1500 * @endcode
1501 *
1502 * The next part is going to be tricky. During execution of refinement, a
1503 * few hp-algorithms need to interfere with the actual refinement process on
1504 * the Triangulation object. We do this by connecting several functions to
1505 * Triangulation::Signals: signals will be called at different stages during
1506 * the actual refinement process and trigger all connected functions. We
1507 * require this functionality for load balancing and to limit the polynomial
1508 * degrees of neighboring cells.
1509 *
1510
1511 *
1512 * For the former, we would like to assign a weight to every cell that is
1513 * proportional to the number of degrees of freedom of its future finite
1514 * element. The library offers a class parallel::CellWeights that allows to
1515 * easily attach individual weights at the right place during the refinement
1516 * process, i.e., after all refine and coarsen flags have been set correctly
1517 * for hp-adaptation and right before repartitioning for load balancing is
1518 * about to happen. Functions can be registered that will attach weights in
1519 * the form that @f$a (n_\text{dofs})^b@f$ with a provided pair of parameters
1520 * @f$(a,b)@f$. We register such a function in the following.
1521 *
1522
1523 *
1524 * For load balancing, efficient solvers like the one we use should scale
1525 * linearly with the number of degrees of freedom owned. We set the
1526 * parameters for cell weighting correspondingly: A weighting factor of @f$1@f$
1527 * and an exponent of @f$1@f$ (see the definitions of the `weighting_factor` and
1528 * `weighting_exponent` above).
1529 *
1530 * @code
1531 *   cell_weights = std::make_unique<parallel::CellWeights<dim>>(
1532 *   dof_handler,
1534 *   {prm.weighting_factor, prm.weighting_exponent}));
1535 *  
1536 * @endcode
1537 *
1538 * In h-adaptive applications, we ensure a 2:1 mesh balance by limiting the
1539 * difference of refinement levels of neighboring cells to one. With the
1540 * second call in the following code snippet, we will ensure the same for
1541 * p-levels on neighboring cells: levels of future finite elements are not
1542 * allowed to differ by more than a specified difference. The function
1543 * hp::Refinement::limit_p_level_difference takes care of this, but needs to
1544 * be connected to a very specific signal in the parallel context. The issue
1545 * is that we need to know how the mesh will be actually refined to set
1546 * future FE indices accordingly. As we ask the p4est oracle to perform
1547 * refinement, we need to ensure that the Triangulation has been updated
1548 * with the adaptation flags of the oracle first. An instantiation of
1550 * that for the duration of its life. Thus, we will create an object of this
1551 * class right before limiting the p-level difference, and connect the
1552 * corresponding lambda function to the signal
1553 * Triangulation::Signals::post_p4est_refinement, which will be triggered
1554 * after the oracle got refined, but before the Triangulation is refined.
1555 * Furthermore, we specify that this function will be connected to the front
1556 * of the signal, to ensure that the modification is performed before any
1557 * other function connected to the same signal.
1558 *
1559 * @code
1561 *   [&, min_fe_index]() {
1563 *   refine_modifier(triangulation);
1565 *   prm.max_p_level_difference,
1566 *   /*contains=*/min_fe_index);
1567 *   },
1568 *   boost::signals2::at_front);
1569 *   }
1570 *  
1571 *  
1572 *  
1573 * @endcode
1574 *
1575 *
1576 * <a name="step_75-LaplaceProbleminitialize_grid"></a>
1577 * <h4>LaplaceProblem::initialize_grid</h4>
1578 *
1579
1580 *
1581 * For a L-shaped domain, we could use the function GridGenerator::hyper_L()
1582 * as demonstrated in @ref step_50 "step-50". However in the 2d case, that particular
1583 * function removes the first quadrant, while we need the fourth quadrant
1584 * removed in our scenario. Thus, we will use a different function
1585 * GridGenerator::subdivided_hyper_L() which gives us more options to create
1586 * the mesh. Furthermore, we formulate that function in a way that it also
1587 * generates a 3d mesh: the 2d L-shaped domain will basically elongated by 1
1588 * in the positive z-direction.
1589 *
1590
1591 *
1592 * We first pretend to build a GridGenerator::subdivided_hyper_rectangle().
1593 * The parameters that we need to provide are Point objects for the lower left
1594 * and top right corners, as well as the number of repetitions that the base
1595 * mesh will have in each direction. We provide them for the first two
1596 * dimensions and treat the higher third dimension separately.
1597 *
1598
1599 *
1600 * To create a L-shaped domain, we need to remove the excess cells. For this,
1601 * we specify the <code>cells_to_remove</code> accordingly. We would like to
1602 * remove one cell in every cell from the negative direction, but remove one
1603 * from the positive x-direction.
1604 *
1605
1606 *
1607 * On the coarse grid, we set the initial active FE indices and distribute the
1608 * degrees of freedom once. We do that in order to assign the hp::FECollection
1609 * to the DoFHandler, so that all cells know how many DoFs they are going to
1610 * have. This step is mandatory for the weighted load balancing algorithm,
1611 * which will be called implicitly in
1612 * parallel::distributed::Triangulation::refine_global().
1613 *
1614 * @code
1615 *   template <int dim>
1616 *   void LaplaceProblem<dim>::initialize_grid()
1617 *   {
1618 *   TimerOutput::Scope t(computing_timer, "initialize grid");
1619 *  
1620 *   std::vector<unsigned int> repetitions(dim);
1621 *   Point<dim> bottom_left, top_right;
1622 *   for (unsigned int d = 0; d < dim; ++d)
1623 *   if (d < 2)
1624 *   {
1625 *   repetitions[d] = 2;
1626 *   bottom_left[d] = -1.;
1627 *   top_right[d] = 1.;
1628 *   }
1629 *   else
1630 *   {
1631 *   repetitions[d] = 1;
1632 *   bottom_left[d] = 0.;
1633 *   top_right[d] = 1.;
1634 *   }
1635 *  
1636 *   std::vector<int> cells_to_remove(dim, 1);
1637 *   cells_to_remove[0] = -1;
1638 *  
1640 *   triangulation, repetitions, bottom_left, top_right, cells_to_remove);
1641 *  
1642 *   const unsigned int min_fe_index = prm.min_p_degree - 1;
1643 *   for (const auto &cell : dof_handler.active_cell_iterators())
1644 *   if (cell->is_locally_owned())
1645 *   cell->set_active_fe_index(min_fe_index);
1646 *  
1647 *   dof_handler.distribute_dofs(fe_collection);
1648 *  
1649 *   triangulation.refine_global(prm.min_h_level);
1650 *   }
1651 *  
1652 *  
1653 *  
1654 * @endcode
1655 *
1656 *
1657 * <a name="step_75-LaplaceProblemsetup_system"></a>
1658 * <h4>LaplaceProblem::setup_system</h4>
1659 *
1660
1661 *
1662 * This function looks exactly the same to the one of @ref step_40 "step-40", but you will
1663 * notice the absence of the system matrix as well as the scaffold that
1664 * surrounds it. Instead, we will initialize the MatrixFree formulation of the
1665 * <code>laplace_operator</code> here. For boundary conditions, we will use
1666 * the Solution class introduced earlier in this tutorial.
1667 *
1668 * @code
1669 *   template <int dim>
1670 *   void LaplaceProblem<dim>::setup_system()
1671 *   {
1672 *   TimerOutput::Scope t(computing_timer, "setup system");
1673 *  
1674 *   dof_handler.distribute_dofs(fe_collection);
1675 *  
1676 *   locally_owned_dofs = dof_handler.locally_owned_dofs();
1677 *   locally_relevant_dofs =
1679 *  
1680 *   locally_relevant_solution.reinit(locally_owned_dofs,
1681 *   locally_relevant_dofs,
1682 *   mpi_communicator);
1683 *   system_rhs.reinit(locally_owned_dofs, mpi_communicator);
1684 *  
1685 *   constraints.clear();
1686 *   constraints.reinit(locally_owned_dofs, locally_relevant_dofs);
1687 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
1689 *   mapping_collection, dof_handler, 0, Solution<dim>(), constraints);
1690 *   constraints.close();
1691 *  
1692 *   laplace_operator.reinit(mapping_collection,
1693 *   dof_handler,
1694 *   quadrature_collection,
1695 *   constraints,
1696 *   system_rhs);
1697 *   }
1698 *  
1699 *  
1700 *  
1701 * @endcode
1702 *
1703 *
1704 * <a name="step_75-LaplaceProblemprint_diagnostics"></a>
1705 * <h4>LaplaceProblem::print_diagnostics</h4>
1706 *
1707
1708 *
1709 * This is a function that prints additional diagnostics about the equation
1710 * system and its partitioning. In addition to the usual global number of
1711 * active cells and degrees of freedom, we also output their local
1712 * equivalents. For a regulated output, we will communicate the local
1713 * quantities with a Utilities::MPI::gather operation to the first process
1714 * which will then output all information. Output of local quantities is
1715 * limited to the first 8 processes to avoid cluttering the terminal.
1716 *
1717
1718 *
1719 * Furthermore, we would like to print the frequencies of the polynomial
1720 * degrees in the numerical discretization. Since this information is only
1721 * stored locally, we will count the finite elements on locally owned cells
1722 * and later communicate them via Utilities::MPI::sum.
1723 *
1724 * @code
1725 *   template <int dim>
1726 *   void LaplaceProblem<dim>::print_diagnostics()
1727 *   {
1728 *   const unsigned int first_n_processes =
1729 *   std::min<unsigned int>(8,
1730 *   Utilities::MPI::n_mpi_processes(mpi_communicator));
1731 *   const bool output_cropped =
1732 *   first_n_processes < Utilities::MPI::n_mpi_processes(mpi_communicator);
1733 *  
1734 *   {
1735 *   pcout << " Number of active cells: "
1736 *   << triangulation.n_global_active_cells() << std::endl
1737 *   << " by partition: ";
1738 *  
1739 *   std::vector<unsigned int> n_active_cells_per_subdomain =
1740 *   Utilities::MPI::gather(mpi_communicator,
1741 *   triangulation.n_locally_owned_active_cells());
1742 *   for (unsigned int i = 0; i < first_n_processes; ++i)
1743 *   pcout << ' ' << n_active_cells_per_subdomain[i];
1744 *   if (output_cropped)
1745 *   pcout << " ...";
1746 *   pcout << std::endl;
1747 *   }
1748 *  
1749 *   {
1750 *   pcout << " Number of degrees of freedom: " << dof_handler.n_dofs()
1751 *   << std::endl
1752 *   << " by partition: ";
1753 *  
1754 *   std::vector<types::global_dof_index> n_dofs_per_subdomain =
1755 *   Utilities::MPI::gather(mpi_communicator,
1756 *   dof_handler.n_locally_owned_dofs());
1757 *   for (unsigned int i = 0; i < first_n_processes; ++i)
1758 *   pcout << ' ' << n_dofs_per_subdomain[i];
1759 *   if (output_cropped)
1760 *   pcout << " ...";
1761 *   pcout << std::endl;
1762 *   }
1763 *  
1764 *   {
1765 *   std::vector<types::global_dof_index> n_constraints_per_subdomain =
1766 *   Utilities::MPI::gather(mpi_communicator, constraints.n_constraints());
1767 *  
1768 *   pcout << " Number of constraints: "
1769 *   << std::accumulate(n_constraints_per_subdomain.begin(),
1770 *   n_constraints_per_subdomain.end(),
1771 *   0)
1772 *   << std::endl
1773 *   << " by partition: ";
1774 *   for (unsigned int i = 0; i < first_n_processes; ++i)
1775 *   pcout << ' ' << n_constraints_per_subdomain[i];
1776 *   if (output_cropped)
1777 *   pcout << " ...";
1778 *   pcout << std::endl;
1779 *   }
1780 *  
1781 *   {
1782 *   std::vector<unsigned int> n_fe_indices(fe_collection.size(), 0);
1783 *   for (const auto &cell : dof_handler.active_cell_iterators())
1784 *   if (cell->is_locally_owned())
1785 *   n_fe_indices[cell->active_fe_index()]++;
1786 *  
1787 *   Utilities::MPI::sum(n_fe_indices, mpi_communicator, n_fe_indices);
1788 *  
1789 *   pcout << " Frequencies of poly. degrees:";
1790 *   for (unsigned int i = 0; i < fe_collection.size(); ++i)
1791 *   if (n_fe_indices[i] > 0)
1792 *   pcout << ' ' << fe_collection[i].degree << ':' << n_fe_indices[i];
1793 *   pcout << std::endl;
1794 *   }
1795 *   }
1796 *  
1797 *  
1798 *  
1799 * @endcode
1800 *
1801 *
1802 * <a name="step_75-LaplaceProblemsolve_system"></a>
1803 * <h4>LaplaceProblem::solve_system</h4>
1804 *
1805
1806 *
1807 * The scaffold around the solution is similar to the one of @ref step_40 "step-40". We
1808 * prepare a vector that matches the requirements of MatrixFree and collect
1809 * the locally-relevant degrees of freedoms we solved the equation system. The
1810 * solution happens with the function introduced earlier.
1811 *
1812 * @code
1813 *   template <int dim>
1814 *   void LaplaceProblem<dim>::solve_system()
1815 *   {
1816 *   TimerOutput::Scope t(computing_timer, "solve system");
1817 *  
1818 *   LinearAlgebra::distributed::Vector<double> completely_distributed_solution;
1819 *   laplace_operator.initialize_dof_vector(completely_distributed_solution);
1820 *  
1821 *   SolverControl solver_control(system_rhs.size(),
1822 *   prm.tolerance_factor * system_rhs.l2_norm());
1823 *  
1824 *   solve_with_gmg(solver_control,
1825 *   laplace_operator,
1826 *   completely_distributed_solution,
1827 *   system_rhs,
1828 *   prm.mg_data,
1829 *   mapping_collection,
1830 *   dof_handler,
1831 *   quadrature_collection);
1832 *  
1833 *   pcout << " Solved in " << solver_control.last_step() << " iterations."
1834 *   << std::endl;
1835 *  
1836 *   constraints.distribute(completely_distributed_solution);
1837 *  
1838 *   locally_relevant_solution.copy_locally_owned_data_from(
1839 *   completely_distributed_solution);
1840 *   locally_relevant_solution.update_ghost_values();
1841 *   }
1842 *  
1843 *  
1844 *  
1845 * @endcode
1846 *
1847 *
1848 * <a name="step_75-LaplaceProblemcompute_indicators"></a>
1849 * <h4>LaplaceProblem::compute_indicators</h4>
1850 *
1851
1852 *
1853 * This function contains only a part of the typical <code>refine_grid</code>
1854 * function from other tutorials and is new in that sense. Here, we will only
1855 * calculate all indicators for adaptation with actually refining the grid. We
1856 * do this for the purpose of writing all indicators to the file system, so we
1857 * store them for later.
1858 *
1859
1860 *
1861 * Since we are dealing the an elliptic problem, we will make use of the
1862 * KellyErrorEstimator again, but with a slight difference. Modifying the
1863 * scaling factor of the underlying face integrals to be dependent on the
1864 * actual polynomial degree of the neighboring elements is favorable in
1865 * hp-adaptive applications @cite davydov2017hp. We can do this by specifying
1866 * the very last parameter from the additional ones you notices. The others
1867 * are actually just the defaults.
1868 *
1869
1870 *
1871 * For the purpose of hp-adaptation, we will calculate smoothness estimates
1872 * with the strategy presented in the tutorial introduction and use the
1873 * implementation in SmoothnessEstimator::Legendre. In the Parameters struct,
1874 * we set the minimal polynomial degree to 2 as it seems that the smoothness
1875 * estimation algorithms have trouble with linear elements.
1876 *
1877 * @code
1878 *   template <int dim>
1879 *   void LaplaceProblem<dim>::compute_indicators()
1880 *   {
1881 *   TimerOutput::Scope t(computing_timer, "compute indicators");
1882 *  
1883 *   estimated_error_per_cell.grow_or_shrink(triangulation.n_active_cells());
1885 *   dof_handler,
1886 *   face_quadrature_collection,
1887 *   std::map<types::boundary_id, const Function<dim> *>(),
1888 *   locally_relevant_solution,
1889 *   estimated_error_per_cell,
1890 *   /*component_mask=*/ComponentMask(),
1891 *   /*coefficients=*/nullptr,
1892 *   /*n_threads=*/numbers::invalid_unsigned_int,
1893 *   /*subdomain_id=*/numbers::invalid_subdomain_id,
1894 *   /*material_id=*/numbers::invalid_material_id,
1895 *   /*strategy=*/
1897 *  
1898 *   hp_decision_indicators.grow_or_shrink(triangulation.n_active_cells());
1900 *   dof_handler,
1901 *   locally_relevant_solution,
1902 *   hp_decision_indicators);
1903 *   }
1904 *  
1905 *  
1906 *  
1907 * @endcode
1908 *
1909 *
1910 * <a name="step_75-LaplaceProblemadapt_resolution"></a>
1911 * <h4>LaplaceProblem::adapt_resolution</h4>
1912 *
1913
1914 *
1915 * With the previously calculated indicators, we will finally flag all cells
1916 * for adaptation and also execute refinement in this function. As in previous
1917 * tutorials, we will use the "fixed number" strategy, but now for
1918 * hp-adaptation.
1919 *
1920 * @code
1921 *   template <int dim>
1922 *   void LaplaceProblem<dim>::adapt_resolution()
1923 *   {
1924 *   TimerOutput::Scope t(computing_timer, "adapt resolution");
1925 *  
1926 * @endcode
1927 *
1928 * First, we will set refine and coarsen flags based on the error estimates
1929 * on each cell. There is nothing new here.
1930 *
1931
1932 *
1933 * We will use general refine and coarsen fractions that have been
1934 * elaborated in the other deal.II tutorials: using the fixed number
1935 * strategy, we will flag 30% of all cells for refinement and 3% for
1936 * coarsening, as provided in the Parameters struct.
1937 *
1938 * @code
1940 *   triangulation,
1941 *   estimated_error_per_cell,
1942 *   prm.refine_fraction,
1943 *   prm.coarsen_fraction);
1944 *  
1945 * @endcode
1946 *
1947 * Next, we will make all adjustments for hp-adaptation. We want to refine
1948 * and coarsen those cells flagged in the previous step, but need to decide
1949 * if we would like to do it by adjusting the grid resolution or the
1950 * polynomial degree.
1951 *
1952
1953 *
1954 * The next function call sets future FE indices according to the previously
1955 * calculated smoothness indicators as p-adaptation indicators. These
1956 * indices will only be set on those cells that have refine or coarsen flags
1957 * assigned.
1958 *
1959
1960 *
1961 * For the p-adaptation fractions, we will take an educated guess. Since we
1962 * only expect a single singularity in our scenario, i.e., in the origin of
1963 * the domain, and a smooth solution anywhere else, we would like to
1964 * strongly prefer to use p-adaptation over h-adaptation. This reflects in
1965 * our choice of a fraction of 90% for both p-refinement and p-coarsening.
1966 *
1967 * @code
1969 *   hp_decision_indicators,
1970 *   prm.p_refine_fraction,
1971 *   prm.p_coarsen_fraction);
1972 *  
1973 * @endcode
1974 *
1975 * After setting all indicators, we will remove those that exceed the
1976 * specified limits of the provided level ranges in the Parameters struct.
1977 * This limitation naturally arises for p-adaptation as the number of
1978 * supplied finite elements is limited. In addition, we registered a custom
1979 * hierarchy for p-adaptation in the constructor. Now, we need to do this
1980 * manually in the h-adaptive context like in @ref step_31 "step-31".
1981 *
1982
1983 *
1984 * We will iterate over all cells on the designated min and max levels and
1985 * remove the corresponding flags. As an alternative, we could also flag
1986 * these cells for p-adaptation by setting future FE indices accordingly
1987 * instead of simply clearing the refine and coarsen flags.
1988 *
1989 * @code
1990 *   Assert(triangulation.n_levels() >= prm.min_h_level + 1 &&
1991 *   triangulation.n_levels() <= prm.max_h_level + 1,
1992 *   ExcInternalError());
1993 *  
1994 *   if (triangulation.n_levels() > prm.max_h_level)
1995 *   for (const auto &cell :
1996 *   triangulation.active_cell_iterators_on_level(prm.max_h_level))
1997 *   cell->clear_refine_flag();
1998 *  
1999 *   for (const auto &cell :
2000 *   triangulation.active_cell_iterators_on_level(prm.min_h_level))
2001 *   cell->clear_coarsen_flag();
2002 *  
2003 * @endcode
2004 *
2005 * At this stage, we have both the future FE indices and the classic refine
2006 * and coarsen flags set. The latter will be interpreted by
2007 * Triangulation::execute_coarsening_and_refinement() for h-adaptation, and
2008 * our previous modification ensures that the resulting Triangulation stays
2009 * within the specified level range.
2010 *
2011
2012 *
2013 * Now, we would like to only impose one type of adaptation on cells, which
2014 * is what the next function will sort out for us. In short, on cells which
2015 * have both types of indicators assigned, we will favor the p-adaptation
2016 * one and remove the h-adaptation one.
2017 *
2018 * @code
2019 *   hp::Refinement::choose_p_over_h(dof_handler);
2020 *  
2021 * @endcode
2022 *
2023 * In the end, we are left to execute coarsening and refinement. Here, not
2024 * only the grid will be updated, but also all previous future FE indices
2025 * will become active.
2026 *
2027
2028 *
2029 * Remember that we have attached functions to triangulation signals in the
2030 * constructor, will be triggered in this function call. So there is even
2031 * more happening: weighted repartitioning will be performed to ensure load
2032 * balancing, as well as we will limit the difference of p-levels between
2033 * neighboring cells.
2034 *
2035 * @code
2036 *   triangulation.execute_coarsening_and_refinement();
2037 *   }
2038 *  
2039 *  
2040 *  
2041 * @endcode
2042 *
2043 *
2044 * <a name="step_75-LaplaceProblemoutput_results"></a>
2045 * <h4>LaplaceProblem::output_results</h4>
2046 *
2047
2048 *
2049 * Writing results to the file system in parallel applications works exactly
2050 * like in @ref step_40 "step-40". In addition to the data containers that we prepared
2051 * throughout the tutorial, we would also like to write out the polynomial
2052 * degree of each finite element on the grid as well as the subdomain each
2053 * cell belongs to. We prepare necessary containers for this in the scope of
2054 * this function.
2055 *
2056 * @code
2057 *   template <int dim>
2058 *   void LaplaceProblem<dim>::output_results(const unsigned int cycle)
2059 *   {
2060 *   TimerOutput::Scope t(computing_timer, "output results");
2061 *  
2062 *   Vector<float> fe_degrees(triangulation.n_active_cells());
2063 *   for (const auto &cell : dof_handler.active_cell_iterators())
2064 *   if (cell->is_locally_owned())
2065 *   fe_degrees(cell->active_cell_index()) = cell->get_fe().degree;
2066 *  
2067 *   Vector<float> subdomain(triangulation.n_active_cells());
2068 *   for (auto &subd : subdomain)
2070 *  
2071 *   DataOut<dim> data_out;
2072 *   data_out.attach_dof_handler(dof_handler);
2073 *   data_out.add_data_vector(locally_relevant_solution, "solution");
2074 *   data_out.add_data_vector(fe_degrees, "fe_degree");
2075 *   data_out.add_data_vector(subdomain, "subdomain");
2076 *   data_out.add_data_vector(estimated_error_per_cell, "error");
2077 *   data_out.add_data_vector(hp_decision_indicators, "hp_indicator");
2078 *   data_out.build_patches(mapping_collection);
2079 *  
2080 *   data_out.write_vtu_with_pvtu_record(
2081 *   "./", "solution", cycle, mpi_communicator, 2, 1);
2082 *   }
2083 *  
2084 *  
2085 *  
2086 * @endcode
2087 *
2088 *
2089 * <a name="step_75-LaplaceProblemrun"></a>
2090 * <h4>LaplaceProblem::run</h4>
2091 *
2092
2093 *
2094 * The actual run function again looks very familiar to @ref step_40 "step-40". The only
2095 * addition is the bracketed section that precedes the actual cycle loop.
2096 * Here, we will pre-calculate the Legendre transformation matrices. In
2097 * general, these will be calculated on the fly via lazy allocation whenever a
2098 * certain matrix is needed. For timing purposes however, we would like to
2099 * calculate them all at once before the actual time measurement begins. We
2100 * will thus designate their calculation to their own scope.
2101 *
2102 * @code
2103 *   template <int dim>
2104 *   void LaplaceProblem<dim>::run()
2105 *   {
2106 *   pcout << "Running with Trilinos on "
2107 *   << Utilities::MPI::n_mpi_processes(mpi_communicator)
2108 *   << " MPI rank(s)..." << std::endl;
2109 *  
2110 *   {
2111 *   pcout << "Calculating transformation matrices..." << std::endl;
2112 *   TimerOutput::Scope t(computing_timer, "calculate transformation");
2113 *   legendre->precalculate_all_transformation_matrices();
2114 *   }
2115 *  
2116 *   for (unsigned int cycle = 0; cycle < prm.n_cycles; ++cycle)
2117 *   {
2118 *   pcout << "Cycle " << cycle << ':' << std::endl;
2119 *  
2120 *   if (cycle == 0)
2121 *   initialize_grid();
2122 *   else
2123 *   adapt_resolution();
2124 *  
2125 *   setup_system();
2126 *  
2127 *   print_diagnostics();
2128 *  
2129 *   solve_system();
2130 *  
2131 *   compute_indicators();
2132 *  
2133 *   if (Utilities::MPI::n_mpi_processes(mpi_communicator) <= 32)
2134 *   output_results(cycle);
2135 *  
2136 *   computing_timer.print_summary();
2137 *   computing_timer.reset();
2138 *  
2139 *   pcout << std::endl;
2140 *   }
2141 *   }
2142 *   } // namespace Step75
2143 *  
2144 *  
2145 *  
2146 * @endcode
2147 *
2148 *
2149 * <a name="step_75-main"></a>
2150 * <h4>main()</h4>
2151 *
2152
2153 *
2154 * The final function is the <code>main</code> function that will ultimately
2155 * create and run a LaplaceOperator instantiation. Its structure is similar to
2156 * most other tutorial programs.
2157 *
2158 * @code
2159 *   int main(int argc, char *argv[])
2160 *   {
2161 *   try
2162 *   {
2163 *   using namespace dealii;
2164 *   using namespace Step75;
2165 *  
2166 *   Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
2167 *  
2168 *   Parameters prm;
2169 *   LaplaceProblem<2> laplace_problem(prm);
2170 *   laplace_problem.run();
2171 *   }
2172 *   catch (std::exception &exc)
2173 *   {
2174 *   std::cerr << std::endl
2175 *   << std::endl
2176 *   << "----------------------------------------------------"
2177 *   << std::endl;
2178 *   std::cerr << "Exception on processing: " << std::endl
2179 *   << exc.what() << std::endl
2180 *   << "Aborting!" << std::endl
2181 *   << "----------------------------------------------------"
2182 *   << std::endl;
2183 *  
2184 *   return 1;
2185 *   }
2186 *   catch (...)
2187 *   {
2188 *   std::cerr << std::endl
2189 *   << std::endl
2190 *   << "----------------------------------------------------"
2191 *   << std::endl;
2192 *   std::cerr << "Unknown exception!" << std::endl
2193 *   << "Aborting!" << std::endl
2194 *   << "----------------------------------------------------"
2195 *   << std::endl;
2196 *   return 1;
2197 *   }
2198 *  
2199 *   return 0;
2200 *   }
2201 * @endcode
2202<a name="step_75-Results"></a><h1>Results</h1>
2203
2204
2205When you run the program with the given parameters on four processes in
2206release mode, your terminal output should look like this:
2207@code
2208Running with Trilinos on 4 MPI rank(s)...
2209Calculating transformation matrices...
2210Cycle 0:
2211 Number of active cells: 3072
2212 by partition: 768 768 768 768
2213 Number of degrees of freedom: 12545
2214 by partition: 3201 3104 3136 3104
2215 Number of constraints: 542
2216 by partition: 165 74 138 165
2217 Frequencies of poly. degrees: 2:3072
2218 Solved in 7 iterations.
2219
2220
2221+---------------------------------------------+------------+------------+
2222| Total wallclock time elapsed since start | 0.172s | |
2223| | | |
2224| Section | no. calls | wall time | % of total |
2225+---------------------------------+-----------+------------+------------+
2226| calculate transformation | 1 | 0.0194s | 11% |
2227| compute indicators | 1 | 0.00676s | 3.9% |
2228| initialize grid | 1 | 0.011s | 6.4% |
2229| output results | 1 | 0.0343s | 20% |
2230| setup system | 1 | 0.00839s | 4.9% |
2231| solve system | 1 | 0.0896s | 52% |
2232+---------------------------------+-----------+------------+------------+
2233
2234
2235Cycle 1:
2236 Number of active cells: 3351
2237 by partition: 875 761 843 872
2238 Number of degrees of freedom: 18228
2239 by partition: 4535 4735 4543 4415
2240 Number of constraints: 1202
2241 by partition: 303 290 326 283
2242 Frequencies of poly. degrees: 2:2522 3:829
2243 Solved in 7 iterations.
2244
2245
2246+---------------------------------------------+------------+------------+
2247| Total wallclock time elapsed since start | 0.165s | |
2248| | | |
2249| Section | no. calls | wall time | % of total |
2250+---------------------------------+-----------+------------+------------+
2251| adapt resolution | 1 | 0.00473s | 2.9% |
2252| compute indicators | 1 | 0.00764s | 4.6% |
2253| output results | 1 | 0.0243s | 15% |
2254| setup system | 1 | 0.00662s | 4% |
2255| solve system | 1 | 0.121s | 74% |
2256+---------------------------------+-----------+------------+------------+
2257
2258
2259...
2260
2261
2262Cycle 7:
2263 Number of active cells: 5610
2264 by partition: 1324 1483 1482 1321
2265 Number of degrees of freedom: 82047
2266 by partition: 21098 19960 20111 20878
2267 Number of constraints: 14357
2268 by partition: 3807 3229 3554 3767
2269 Frequencies of poly. degrees: 2:1126 3:1289 4:2725 5:465 6:5
2270 Solved in 7 iterations.
2271
2272
2273+---------------------------------------------+------------+------------+
2274| Total wallclock time elapsed since start | 1.83s | |
2275| | | |
2276| Section | no. calls | wall time | % of total |
2277+---------------------------------+-----------+------------+------------+
2278| adapt resolution | 1 | 0.00834s | 0.46% |
2279| compute indicators | 1 | 0.0178s | 0.97% |
2280| output results | 1 | 0.0434s | 2.4% |
2281| setup system | 1 | 0.0165s | 0.9% |
2282| solve system | 1 | 1.74s | 95% |
2283+---------------------------------+-----------+------------+------------+
2284@endcode
2285
2286When running the code with more processes, you will notice slight
2287differences in the number of active cells and degrees of freedom. This
2288is due to the fact that solver and preconditioner depend on the
2289partitioning of the problem, which might yield to slight differences of
2290the solution in the last digits and ultimately yields to different
2291adaptation behavior.
2292
2293Furthermore, the number of iterations for the solver stays about the
2294same in all cycles despite hp-adaptation, indicating the robustness of
2295the proposed algorithms and promising good scalability for even larger
2296problem sizes and on more processes.
2297
2298Let us have a look at the graphical output of the program. After all
2299refinement cycles in the given parameter configuration, the actual
2300discretized function space looks like the following with its
2301partitioning on twelve processes on the left and the polynomial degrees
2302of finite elements on the right. In the left picture, each color
2303represents a unique subdomain. In the right picture, the lightest color
2304corresponds to the polynomial degree two and the darkest one corresponds
2305to degree six:
2306
2307<div class="twocolumn" style="width: 80%; text-align: center;">
2308 <div>
2309 <img src="https://www.dealii.org/images/steps/developer/step-75.subdomains-07.svg"
2310 alt="Partitioning after seven refinements.">
2311 </div>
2312 <div>
2313 <img src="https://www.dealii.org/images/steps/developer/step-75.fedegrees-07.svg"
2314 alt="Local approximation degrees after seven refinements.">
2315 </div>
2316</div>
2317
2318
2319
2320<a name="step-75-extensions"></a>
2321<a name="step_75-Possibilitiesforextensions"></a><h3>Possibilities for extensions</h3>
2322
2323
2324This tutorial shows only one particular way how to use parallel
2325hp-adaptive finite element methods. In the following paragraphs, you
2326will get to know which alternatives are possible. Most of these
2327extensions are already part of https://github.com/marcfehling/hpbox/,
2328which provides you with implementation examples that you can play
2329around with.
2330
2331
2332<a name="step_75-Differenthpdecisionstrategies"></a><h4>Different hp-decision strategies</h4>
2333
2334
2335The deal.II library offers multiple strategies to decide which type of
2336adaptation to impose on cells: either adjust the grid resolution or
2337change the polynomial degree. We only presented the <i>Legendre
2338coefficient decay</i> strategy in this tutorial, while @ref step_27 "step-27"
2339demonstrated the <i>Fourier</i> equivalent of the same idea.
2340
2341See the "possibilities for extensions" section of @ref step_27 "step-27" for an
2342overview over these strategies, or the corresponding documentation
2343for a detailed description.
2344
2345There, another strategy is mentioned that has not been shown in any
2346tutorial so far: the strategy based on <i>refinement history</i>. The
2347usage of this method for parallel distributed applications is more
2348tricky than the others, so we will highlight the challenges that come
2349along with it. We need information about the final state of refinement
2350flags, and we need to transfer the solution across refined meshes. For
2351the former, we need to attach the hp::Refinement::predict_error()
2352function to the Triangulation::Signals::post_p4est_refinement signal in
2353a way that it will be called <i>after</i> the
2354hp::Refinement::limit_p_level_difference() function. At this stage, all
2355refinement flags and future FE indices are terminally set and a reliable
2356prediction of the error is possible. The predicted error then needs to
2357be transferred across refined meshes with the aid of
2358parallel::distributed::CellDataTransfer.
2359
2360Try implementing one of these strategies into this tutorial and observe
2361the subtle changes to the results. You will notice that all strategies
2362are capable of identifying the singularities near the reentrant corners
2363and will perform @f$h@f$-refinement in these regions, while preferring
2364@f$p@f$-refinement in the bulk domain. A detailed comparison of these
2365strategies is presented in @cite fehling2020 .
2366
2367
2368<a name="step_75-Solvewithmatrixbasedmethods"></a><h4>Solve with matrix-based methods</h4>
2369
2370
2371This tutorial focuses solely on matrix-free strategies. All hp-adaptive
2372algorithms however also work with matrix-based approaches in the
2373parallel distributed context.
2374
2375To create a system matrix, you can either use the
2376LaplaceOperator::get_system_matrix() function, or use an
2377<code>assemble_system()</code> function similar to the one of @ref step_27 "step-27".
2378You can then pass the system matrix to the solver as usual.
2379
2380You can time the results of both matrix-based and matrix-free
2381implementations, quantify the speed-up, and convince yourself which
2382variant is faster.
2383
2384
2385<a name="step_75-Multigridvariants"></a><h4>Multigrid variants</h4>
2386
2387
2388For sake of simplicity, we have restricted ourselves to a single type of
2389coarse-grid solver (CG with AMG), smoother (Chebyshev smoother with
2390point Jacobi preconditioner), and geometric-coarsening scheme (global
2391coarsening) within the multigrid algorithm. Feel free to try out
2392alternatives and investigate their performance and robustness.
2393 *
2394 *
2395<a name="step_75-PlainProg"></a>
2396<h1> The plain program</h1>
2397@include "step-75.cc"
2398*/
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
void reinit(const Triangulation< dim, spacedim > &tria)
void evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
Definition fe_q.h:554
static void estimate(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim - 1 > &quadrature, const std::map< types::boundary_id, const Function< spacedim, Number > * > &neumann_bc, const ReadVector< Number > &solution, Vector< float > &error, const ComponentMask &component_mask={}, const Function< spacedim > *coefficients=nullptr, const unsigned int n_threads=numbers::invalid_unsigned_int, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id, const types::material_id material_id=numbers::invalid_material_id, const Strategy strategy=cell_diameter_over_24)
void initialize(const MGLevelObject< MatrixType2 > &matrices, const typename PreconditionerType::AdditionalData &additional_data=typename PreconditionerType::AdditionalData())
void initialize_dof_vector(VectorType &vec, const unsigned int dof_handler_index=0) const
void reinit(const MappingType &mapping, const DoFHandler< dim > &dof_handler, const AffineConstraints< number2 > &constraint, const QuadratureType &quad, const AdditionalData &additional_data=AdditionalData())
Definition point.h:111
void solve(const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)
@ wall_times
Definition timer.h:651
virtual types::subdomain_id locally_owned_subdomain() const
void coarsen_global(const unsigned int times=1)
virtual void execute_coarsening_and_refinement()
Signals signals
Definition tria.h:2524
unsigned int size() const
Definition collection.h:308
static WeightingFunction ndofs_weighting(const std::pair< float, float > &coefficients)
Point< 2 > second
Definition grid_out.cc:4624
Point< 2 > first
Definition grid_out.cc:4623
unsigned int level
Definition grid_out.cc:4626
IteratorRange< active_cell_iterator > active_cell_iterators() const
#define Assert(cond, exc)
#define AssertThrow(cond, exc)
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:564
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
void make_sparsity_pattern(const DoFHandler< dim, spacedim > &dof_handler, SparsityPatternBase &sparsity_pattern, const AffineConstraints< number > &constraints={}, const bool keep_constrained_dofs=true, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id)
@ update_gradients
Shape function gradients.
#define DEAL_II_NOT_IMPLEMENTED()
IndexSet extract_locally_relevant_dofs(const DoFHandler< dim, spacedim > &dof_handler)
std::array< double, dim > to_spherical(const Point< dim > &point)
void hyper_L(Triangulation< dim > &tria, const double left=-1., const double right=1., const bool colorize=false)
void subdivided_hyper_L(Triangulation< dim, spacedim > &tria, const std::vector< unsigned int > &repetitions, const Point< dim > &bottom_left, const Point< dim > &top_right, const std::vector< int > &n_cells_to_remove)
void refine(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold, const unsigned int max_to_mark=numbers::invalid_unsigned_int)
void coarsen(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold)
void scale(const double scaling_factor, Triangulation< dim, spacedim > &triangulation)
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
Tpetra::Vector< Number, LO, GO, NodeType< MemorySpace > > VectorType
std::vector< std::shared_ptr< const Triangulation< dim, spacedim > > > create_geometric_coarsening_sequence(const Triangulation< dim, spacedim > &tria)
unsigned int create_next_polynomial_coarsening_degree(const unsigned int degree, const PolynomialCoarseningSequenceType &p_sequence)
std::vector< unsigned int > create_polynomial_coarsening_sequence(const unsigned int max_degree, const PolynomialCoarseningSequenceType &p_sequence)
void compute_diagonal(const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free, VectorType &diagonal_global, const std::function< void(FEEvaluation< dim, fe_degree, n_q_points_1d, n_components, Number, VectorizedArrayType > &)> &cell_operation, const unsigned int dof_no=0, const unsigned int quad_no=0, const unsigned int first_selected_component=0, const unsigned int first_vector_component=0)
void compute_matrix(const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free, const AffineConstraints< Number > &constraints, MatrixType &matrix, const std::function< void(FEEvaluation< dim, fe_degree, n_q_points_1d, n_components, Number, VectorizedArrayType > &)> &cell_operation, const unsigned int dof_no=0, const unsigned int quad_no=0, const unsigned int first_selected_component=0)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
Tensor< 2, dim, Number > l(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
SymmetricTensor< 2, dim, Number > b(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
FESeries::Legendre< dim, spacedim > default_fe_series(const hp::FECollection< dim, spacedim > &fe_collection, const unsigned int component=numbers::invalid_unsigned_int)
void coefficient_decay(FESeries::Legendre< dim, spacedim > &fe_legendre, const DoFHandler< dim, spacedim > &dof_handler, const VectorType &solution, Vector< float > &smoothness_indicators, const VectorTools::NormType regression_strategy=VectorTools::Linfty_norm, const double smallest_abs_coefficient=1e-10, const bool only_flagged_cells=false)
void partition(const SparsityPattern &sparsity_pattern, const unsigned int n_partitions, std::vector< unsigned int > &partition_indices, const Partitioner partitioner=Partitioner::metis)
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned int n_mpi_processes(const MPI_Comm mpi_communicator)
Definition mpi.cc:92
T max(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
Definition mpi.cc:107
std::vector< T > gather(const MPI_Comm comm, const T &object_to_send, const unsigned int root_process=0)
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask={})
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
bool limit_p_level_difference(const DoFHandler< dim, spacedim > &dof_handler, const unsigned int max_difference=1, const unsigned int contains_fe_index=0)
void predict_error(const DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &error_indicators, Vector< Number > &predicted_errors, const double gamma_p=std::sqrt(0.4), const double gamma_h=2., const double gamma_n=1.)
void p_adaptivity_fixed_number(const DoFHandler< dim, spacedim > &dof_handler, const Vector< Number > &criteria, const double p_refine_fraction=0.5, const double p_coarsen_fraction=0.5, const ComparisonFunction< std_cxx20::type_identity_t< Number > > &compare_refine=std::greater_equal< Number >(), const ComparisonFunction< std_cxx20::type_identity_t< Number > > &compare_coarsen=std::less_equal< Number >())
Definition hp.h:117
int(&) functions(const void *v1, const void *v2)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
Definition mg.h:81
const types::material_id invalid_material_id
Definition types.h:277
const types::subdomain_id invalid_subdomain_id
Definition types.h:341
static const unsigned int invalid_unsigned_int
Definition types.h:220
void refine_and_coarsen_fixed_number(::Triangulation< dim, spacedim > &tria, const ::Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const types::global_cell_index max_n_cells=std::numeric_limits< types::global_cell_index >::max())
double legendre(unsigned int l, double x)
Definition cmath.h:65
STL namespace.
::VectorizedArray< Number, width > sin(const ::VectorizedArray< Number, width > &)
::VectorizedArray< Number, width > pow(const ::VectorizedArray< Number, width > &, const Number p)
::VectorizedArray< Number, width > abs(const ::VectorizedArray< Number, width > &)
Definition types.h:32
unsigned short int fe_index
Definition types.h:59
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
boost::signals2::signal< void()> post_p4est_refinement
Definition tria.h:2459